An open problem posed there was solved. Added a footnote explaining this
发表信息：
Selection Principles and Covering Properties in Topology (L. Kovinac, editor), Quaderni di Matematica 18, Seconda Universita di Napoli, Caserta 2006, 245-282
The_additivity_number_ of a topological property (relative to a given space) is the minimal number of subspaces with this property whose union does not have the property. The most well-known case is where this number is greater than Aleph_0, i.e. the property is sigma-additive. We give a rather complete survey of the known results about the additivity numbers of a variety of topological covering properties, including those appearing in the Scheepers diagram (which contains, among others, the classical properties of Menger, Hurewicz, Rothberger, and Gerlits-Nagy). Some of the results proved here were not published beforehand, and many open problems are posed.
We establish a surprising connection between Menger's classical covering property and Blass's modern combinatorial notion of groupwise density. This connection implies a short proof of the groupwise density bound on the additivity number for Menger's property.
We show that countable increasing unions preserve a large family of well-studied covering properties, which are not necessarily sigma-additive. Using this, together with infinite-combinatorial methods and simple forcing theoretic methods, we explain several phenomena, settle problems of Just, Miller, Scheepers and Szeptycki [COC2], Gruenhage and Szeptycki [FUfin], Tsaban and Zdomskyy [SFT], and Tsaban [o-bdd, OPiT], and construct topological groups with very strong combinatorial properties.
We answer a question of Just, Miller, Scheepers and Szeptycki whether certain diagonalization properties for sequences of open covers are provably closed under taking finite or countable unions. This is a very concise paper. For a self-contained, complete, and extended treatment of the topic see math.GN/0604451
Motivated by the minimal tower problem, an earlier work studied diagonalizations of covers where the covers are related to linear quasiorders (tau-covers). We deal with two types of combinatorial questions which arise from this study. 1. Two new cardinals introduced in the topological study are expressed in terms of well known cardinals characteristics of the continuum. 2. We study the additivity numbers of the combinatorial notions corresponding to the topological diagonalization notions. This gives new insights on the structure of the eventual dominance ordering on the Baire space, the almos…
We study the preservation of selective covering properties, including classic ones introduced by Menger, Hurewicz, Rothberger, Gerlits and Nagy, and others, under products with some major families of concentrated sets of reals. Our methods include the projection method introduced by the authors in an earlier work, as well as several new methods. Some special consequences of our main results are (definitions provided in the paper): \be \item Every product of a concentrated space with a Hurewicz $\sone(\Ga,\Op)$ space satisfies $\sone(\Ga,\Op)$. On the other hand, assuming \CH{}, for each Sierpi…
We develop the semifilter approach to the classical Menger and Hurewicz covering properties and show that the small cardinal g is a lower bound of the additivity number of the family of Menger subspaces of the Baire space, and under u< g every subset X of the real line with the property Split(Lambda,Lambda) is Hurewicz.
Almost-Uniform Bayesian Convergence to the Truth Is Not Characterized by Countable Additivity on Conditional Hitting Times
The Johns Hopkins University（约翰斯·霍普金斯大学）；The City College of New York & The Graduate Center, The City University of New York（纽约市立学院及纽约市立大学研究生院）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
We show that not every family of generalized microscopic sets forms an ideal. Moreover, we prove that some of these families have some weaker additivity properties and some of them do not have even that.
Iwasa investigated the preservation of various covering properties of opological spaces under Cohen forcing. By improving the argument in Iwasa's paper, we prove that the Rothberger property, the Menger property and selective screenability are also preserved under Cohen forcing and forcing with the measure algebra.
We prove that the Hurewicz property is not preserved by finite products in the Miller model. This is a consequence of the fact that Miller forcing preserves ground model $γ$-spaces.
Which Isbell--Mrówka spaces ($Ψ$-spaces) satisfy the star version of Menger's and Hurewicz's covering properties? Following Bonanzinga and Matveev, this question is considered here from a combinatorial point of view. An example of a $Ψ$-space that is (strongly) star-Menger but not star-Hurewicz is obtained. The PCF-theory function $κ\mapsto\cof([κ]^\alephes)$ is a key tool. Using the method of forcing, a complete answer to a question of Bonanzinga and Matveev is provided. The results also apply to the mentioned covering properties in the realm of Pixley--Roy spaces, to the extent of spaces wit…
We prove that CH implies the existence of a Cohen-indestructible mad family such that the Mathias forcing associated to its filter adds dominating reals, while $\mathfrak b=\mathfrak c$ is consistent with the negation of this statement as witnessed by the Laver model for the consistency of Borel's conjecture.
We give topological characterizations of filters $F$ on $w$ such that the Mathias forcing $M_F$ adds no dominating reals or preserves ground model unbounded families. This allows us to answer some questions of Brendle, Guzmán, Hrušák, Mart\'ınez, Minami, and Tsaban.
No convenient internal characterization of spaces that are productively Lindelof is known. Perhaps the best general result known is Alster's internal characterization, under the Continuum Hypothesis, of productively Lindelof spaces which have a basis of cardinality at most $\aleph_1$. It turns out that topological spaces having Alster's property are also productively weakly Lindelof. The weakly Lindelof spaces form a much larger class of spaces than the Lindelof spaces. In many instances spaces having Alster's property satisfy a seemingly stronger version of Alster's property and consequently …
Recently, Mršević and Reilly discussed some covering properties of a topological space and its associated $α$-topology in both topological and bitopological ways. The main aim of this paper is to investigate some common and controversial covering properties of $\cal T$ and ${\cal T}^α$.
We investigate star-covering properties of $Ψ$-like spaces. We show star-Lindelöfness is reflected by open perfect mappings. In addition, we offer a new equivalence of CH.
Combinatorial covering properties in an uncountable setting: canonical examples
Bar-Ilan University（巴伊兰大学）；Ariel University（阿里埃勒大学）；Lodz University of Technology（罗兹理工大学）；University of Warsaw（华沙大学）；Technische Universität Wien（维也纳工业大学）
A subset of the Cantor cube is null-additive if its algebraic sum with any null set is null. We construct a set of cardinality continuum such that: all continuous images of the set into the Cantor cube are null-additive, it contains a homeomorphic copy of a set that is not null-additive, and it has the property $γ$, a strong combinatorial covering property. We also construct a nontrivial subset of the Cantor cube with the property $γ$ that is not null additive. Set-theoretic assumptions used in our constructions are far milder than used earlier by Galvin--Miller and Bartoszyński--Recław, to ob…
We study productive properties of gamma spaces, and their relation to other, classic and modern, selective covering properties. Among other things, we prove the following results: 1. Solving a problem of F. Jordan, we show that for every unbounded tower set of reals X of cardinality aleph_1, the space Cp(X) is productively FU. In particular, the set X is productively gamma. 2. Solving problems of Scheepers and Weiss, and proving a conjecture of Babinkostova-Scheepers, we prove that, assuming CH, there are gamma spaces whose product is not even Menger. 3. Solving a problem of Scheepers-Tall, we…
In this paper, we continue to investigate topological properties of $\mathcal{I}H$ and its two star versions namely $SS \mathcal{I} H$ and $S \mathcal{I} H$. We characterized $\mathcal{I}$-Hurewicz property by $\mathcal{I}$-Hurewicz Basis property and $\mathcal{I}$-Hurewicz measure zero property for metrizable spaces. We also characterized $\mathcal{I}$-Hurewicz property, star-$\mathcal{I}$-Hurewicz property and strongly star-$\mathcal{I}$-Hurewicz property using selection principles.
Lindelöf spaces are studied in any basic Topology course. However, there are other interesting covering properties with similar behaviour, such as almost Lindelöf, weakly Lindelöf, and quasi-Lindelöf, that have been considered in various research papers. Here we present a comparison between the standard results on Lindelöf spaces and analogous results for weakly and almost Lindelöf spaces. Some theorems, similar to the published ones, will be proved. We also consider counterexamples, most of which have not been included in the standard Topological textbooks, that show the interrelations betwee…
Using the idea of strong uniform convergence on bornology, Caserta, Di Maio and Kočinac studied open covers and selection principles in the realm of metric spaces (associated with a bornology) and function spaces (w.r.t. the topology of strong uniform convergence). We primarily continue in the line initiated before and investigate the behaviour of various selection principles related to these classes of bornological covers. In the process we obtain implications among these selection principles resulting in Scheepers' like diagrams. We also introduce the notion of strong-$\mathfrac{B}$-Hurewicz…
We investigate game-theoretic properties of selection principles related to weaker forms of the Menger and Rothberger properties. For appropriate spaces some of these selection principles are characterized in terms of a corresponding game. We use generic extensions by Cohen reals to illustrate the necessity of some of the hypotheses in our theorems.
Let ${\mathcal A}\subset {\mathcal P}(X)$, $\emptyset, X\in {\mathcal A}$, ${\mathcal A}$ being closed under finite intersections. If $ψ={o},ω,γ$, then $Ψ({\mathcal A})$ is the family of those $ψ$-covers ${\mathcal U}$ for which ${\mathcal U}\subseteq {\mathcal A}$. In~\cite{BL2} I have introduced properties $(Ψ_0$ of a~family $F\subseteq {}^XR$ of real functions. The main result of the paper Theorem reads as follows: if~$Φ=Ω,Γ$, then for any couple $\langle Φ,Ψ\rangle$ different from $\langle Ω,{\mathcal O}\rangle$, $X$ has the covering property~{\rm S}${}_1(Φ({\mathcal A}),Ψ({\mathcal A}))$ …
We study pullback from a topological viewpoint with emphasis on pullback of covering maps. We generalize a triad of Quillen on properties of the pullback functor.
